If you have eight one kilogram blocks, how may different combinations are possible so that the centre of
mass comes to the centre of the square ABCD. All blocks have to be used in each combination. The blocks can be kept only at the corners of the square.
Answers
Answer:
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Explanation:
We can start by drawing a square, or just its vertices on a cartesian coordinate plane (the standard xy-plane). I'll assign the coordinates (0,1), (1,0),(0,-1),(-1,0) to represent the 4 corners / vertices of the square. When these points in the plane are connected by line segments ( but not along the axes ), one has drawn a square with its center at the origin (0,0).
The problem of placing all 8 1kg blocks on the 4 corners in a way that causes the center of mass of the blocks to coincide with the origin is just a matter of “balance", in their distribution.
Along each axis,( x and y ), there must be an equal number of blocks on the opposing corners of the square (The diagonals of the square fall along the x and y axes, here). That is, the same number of blocks will be placed on (1,0), as on (-1,0), and the remaining blocks will be placed on (0,-1) and (0,1), divided equally between the two points.
That gives 4 against 4, along x-axis, with no blocks placed on the y-axis; 3 against 3, with 1 against 1 along y-axis; 2 against 2 on both axes; 1 against 1, along x-axis and 3 against 3 on the y-axis; and 0 against 0 on the x-axis and 4 against 4 on the y-axis. If you distinguish between the 2 diagonals of the square, this gives a total of 5 possible combinations. If not, then only 3 combinations.
Of course, this is all based on the formulas for calculating the position of the center of mass of a system of particles or bodies.
X =( M1*x1 + M2*x2 + ….)/(M1 + M2 +…); and
Y = (M1*y1 + M2*y2 + …)/(M1 + M2 + …).
Basically, one is calculating the average location of matter in a distribution of mass.